Jonathan Smith, on the run from Ohio, was so desperate to come clean that he grabbed the first priest he could find at a Midtown church and copped to bludgeoning his girlfriend 500 miles away, sources told The Post.

“Someone is dead, and I feel really bad about it,” Smith, 48, told the startled clergyman at the Church of St. Francis of Assisi on West 31st Street, according to a law-enforcement source.

“He just grabbed a priest and said, ‘I want to tell you something,’ ” said another priest at the church, who asked to remain anonymous.

The priest whom Smith approached quickly contacted the Midtown South precinct house, and detectives took the suspect’s more formal confession.